Chapter Polynomials and Rationals Review
Chapter
Chapter 2
Section
Chapter Polynomials and Rationals Review
Solutions 59 Videos

Simplify.

(7x^2 - 2x + 1)+(9x^2 -4x + 5)-(4x^2 + 6x - 7)

0.44mins
Q1a

Simplify.

(7a^2 -4ab + 9b^2)-(-a^2+2ab + 6b^2)

0.37mins
Q1b

Determine two non-equivalent polynomials f(x) and g(x), such that f(0) =g(0) and f(1) = g(1).

2.23mins
Q2

Ms. Frizzle has three daughters: Allison, Belle, and Claire. Today, January 1, their ages are, respectively,

 \displaystyle \begin{array}{cccccc} & A(n) = -(n + 30)+(2n + 5) \\ & B(n) = (7 -n)-(32 -2n) \\ & C(n) = (n - 26)-(n + 4) + (n - 3) \\ \end{array} 

All ages are expressed in years, and n represents Ms. Frizzle's age.

a) Are the daughter triples? Explain.

b) Are any of them twins? Explain.

1.05mins
Q3ab

Ms. Frizzle has three daughters: Allison, Belle, and Claire. Today, January 1, their ages are, respectively,

 \displaystyle \begin{array}{cccccc} & A(n) = -(n + 30)+(2n + 5) \\ & B(n) = (7 -n)-(32 -2n) \\ & C(n) = (n - 26)-(n + 4) + (n - 3) \\ \end{array} 

All ages are expressed in years, and n represents Ms. Flangan's age.

(c) How old was Ms. Flanagan when Cassandra was born?

0.33mins
Q3c

Expand and simplify.

-3(7x -5)(4x - 7)

0.59mins
Q4a

Expand and simplify.

-(y^2-4y + 7)(3y^2-5y-3)

1.45mins
Q4b

Expand and simplify.

2(a +b)^3

1.21mins
Q4c

Expand and simplify.

3(x^2 -2)^2(2x - 3)^2

2.38mins
Q4d

The volume of a cone is given by \displaystyle V = \frac{1}{3}\pi r^2h. Determine the volume of the cone in simplified form if the radius is increased by x and the height is increased by 2x.

2.42mins
Q5

Simplify.

(2x^4-3x^2 - 6)+(6x^4 -x^3+4x^2 + 5)

0.38mins
Q6a

Simplify.

(x^2-4)(2x^2+ 5x - 2)

0.35mins
Q6b

Simplify.

-7x(3x^3-7x + 2)-3x(2x^2-5x+ 6)

0.43mins
Q6c

Simplify.

-2x^2(3x^3-7x+2)-x^3(5x^3+2x -8)

0.50mins
Q6d

Simplify.

-2x[5x - (2x -7)] + 6x[3x-(1 + 2x)]

0.45mins
Q6e

Simplify.

(x + 2)^2(x- 1)^2 - (x-4)^2(x + 4)^2

2.25mins
Q6f

Simplify.

(x^2+5x - 3)^2

1.21mins
Q6g

Factor.

12m^2n^3 + 18m^3n^2

0.30mins
Q7a

Factor.

x^2-9x+20

0.25mins
Q7b

Factor.

3x^2+24x + 45

0.30mins
Q7c

Factor.

50x^2-72

0.34mins
Q7d

Factor.

9x^2-6x + 1

0.11mins
Q7e

Factor.

10a^2+a - 3

0.21mins
Q7f

Factor.

2x^2y^4-6x^5y^3+8x^3y

0.33mins
Q8a

Factor.

2x(x + 4)+3(x + 4)

0.12mins
Q8b

Factor.

x^2 -3x-10

0.12mins
Q8c

Factor.

15x^2-53x +42

1.20mins
Q8d

Factor.

a^4 -16

0.25mins
Q8e

Factor.

(m - n)^2-(2m + 3n)^2

0.59mins
Q8f

Simplify. State any restrictions on the variables.

\displaystyle \frac{10a^2b + 15bc^2}{-5b}

0.49mins
Q9a

Simplify. State any restrictions on the variables.

\displaystyle \frac{130x^2y^3-20x^2z^2+50x^2}{10x^2}

0.37mins
Q9b

Simplify. State any restrictions on the variables.

\displaystyle \frac{xy - xyz}{xy}

0.23mins
Q9c

Simplify. State any restrictions on the variables.

\displaystyle \frac{16mnr-24mnp+40kmn}{8mn}

0.42mins
Q9d

Simplify. State any restrictions on the variables.

\displaystyle 8xy^2 + 12x^2y - \frac{6x^3}{2xy}

1.19mins
Q10a

Simplify. State any restrictions on the variables.

\displaystyle \frac{7a-14b}{2(a - 2b)}

0.26mins
Q10b

Simplify. State any restrictions on the variables.

\displaystyle \frac{m + 3}{m^2+10m + 21}

0.26mins
Q10c

Simplify. State any restrictions on the variables.

\displaystyle \frac{4x^2-4x -3}{4x^2-9}

0.42mins
Q10d

Simplify. State any restrictions on the variables.

\displaystyle \frac{3x^2-21x}{7x^2-28x + 21}

0.38mins
Q10e

Simplify. State any restrictions on the variables.

\displaystyle \frac{3x^2-2xy-y^2}{3x^2+4xy+y^2}

0.47mins
Q10f

Simplify. State any restrictions on the variables.

\displaystyle \frac{6x}{8y}\times \frac{2y^2}{3x}

0.31mins
Q12a

Simplify. State any restrictions on the variables.

\displaystyle \frac{10m^2}{3n}\times \frac{6mn}{20m^2}

0.26mins
Q12b

Simplify. State any restrictions on the variables.

\displaystyle \frac{2ab}{5bc} \div\frac{6ac}{10b}

1.02mins
Q12c

Simplify. State any restrictions on the variables.

\displaystyle \frac{5p}{8pq} \div \frac{3p}{12q}

0.43mins
Q12d

Simplify. State any restrictions on the variables.

\displaystyle \frac{x^2}{2xy}\times \frac{x}{2y^2} \div \frac{(3x)^2}{xy^2}

0.54mins
Q13a

Simplify. State any restrictions on the variables.

\displaystyle \frac{x^2-5x + 6}{x^2-1} \times \frac{x^2-4x -5}{x^2-4} \div \frac{x-5}{x^2+3x+2}

1.45mins
Q13b

Simplify. State any restrictions on the variables.

\displaystyle \frac{1-x^2}{1+y} \times \frac{1-y^2}{x + x^2} \div \frac{y^3-y}{x^2}

2.00mins
Q13c

Simplify. State any restrictions on the variables.

\displaystyle \frac{x^2-y^2}{4x^2-y^2}\times \frac{4x^2+8xy+3y^2}{x + y}\div \frac{2x + 3y}{2x -y}

1.47mins
Q13d

Simplify. State any restrictions on the variables.

\displaystyle \frac{4}{5x}-\frac{2}{3x}

0.25mins
Q14a

Simplify. State any restrictions on the variables.

\displaystyle \frac{5}{x + 1}- \frac{2}{x -1}

0.47mins
Q14b

Simplify. State any restrictions on the variables.

\displaystyle \frac{1}{x^2+3x -4} + \frac{1}{x^2+x-12}

0.58mins
Q14c

Simplify. State any restrictions on the variables.

\displaystyle \frac{1}{x^2-5x+6} - \frac{1}{x^2 - 9}

1.24mins
Q14d

Simplify and state any restrictions on the variables.

\displaystyle \frac{1}{2x} - \frac{7}{3x^2}+ \frac{4}{x^3} 

1.11mins
Q15a

Simplify and state any restrictions on the variables.

\displaystyle \frac{3x}{x + 2} + \frac{4x}{x -6} 

0.53mins
Q15b

Simplify and state any restrictions on the variables.

\displaystyle \frac{6x}{x^2 - 5x + 6} - \frac{3x}{x^2 + x - 12} 

Q15c

Simplify and state any restrictions on the variables.

\displaystyle \frac{2(x-2)^2}{x^2 + 6x + 5} \times \frac{3x + 15}{(2- x)^2} 

Q15d

Simplify and state any restrictions on the variables.

\displaystyle \frac{(x - 2y)^2}{x^2 -y^2} \div \frac{(x- 2y)(x + 3y)}{(x + y)^2} 

2.02mins
Q15e

Simplify and state any restrictions on the variables.

\displaystyle \frac{2b -5}{b^2 -2b -15} + \frac{3b}{b^2 + b - 30} \times \frac{b^2 + 8b + 12}{b + 3} 

2.02mins
Q15f

Fred’s final mark in an online course was determined entirely by two exams. The mid-term exam was out of x marks and was worth 25% of his final mark. The final exam was out of 2x marks and was worth 75% of his final mark. Fred scored 40 marks on the first exam and 60 marks on the second exam. Determine the value of x if Fred earned a final mark of 50% in the course.

3.27mins
Q16

Sam plays a game in which he selects three different numbers from 1 to n (n > 3). After he selects his numbers, four different winning numbers from 1 to n are chosen, one at a time. Sam wins if all three of his numbers are among the four winning numbers.

The first number chosen is one of Sam's! His probability of winning is now given by

\displaystyle P(n) = \frac{24}{n^3 - 3n^2 + 2n} \div \frac{3}{n} 

a) Simplify P(n) and state restrictions on n.

b) What would Sam's probability of winning be if

• i) n = 5?
• ii) n = 4?
3.09mins
Q17
Product of Rational Expressions 6 Videos

## Introduction to Rational Expressions

Definition Rational number is any number that can be expressed as \frac{a}{b} where a, b, are integers.

ex \dfrac{x + 1}{x - 1}, \dfrac{x^2 - 2x - 3}{(x + 2)(x - 5)}

3.50mins
Introduction to Rational Expressions

## Simplifying Rational Expressions

ex Simplify & state restrictions

\dfrac{AB}{BC} \times \dfrac{C}{D} = \dfrac{A}{D}, B \neq 0, C \neq 0

2.37mins
Simplifying Rational Expressions

## Simplifying Rational Expression in Factored Form

ex Simplify and state the restrictions.

\dfrac{(2x -1)(x + 5)}{(x -1)(x + 5)}

= \dfrac{2x - 1}{x - 1}, x \neq 1, -5

0.40mins
Simplifying Rational Expressions and its restrictions

ex Simplify and state the restrictions.

\dfrac{2x^2 -x -3}{x^2 + 2x +1}

= \dfrac{2x - 3}{x + 1}, x \neq -1

1.04mins
Simplifying Rational Expressions by Factoring and its Restrictions

## Multiplying Rational Expressions

ex Simplify and state the restrictions.

\dfrac{x^2 -4}{(x + 6)^2} \cdot \frac{x^2 + 9x + 18}{2(2 -x)}

= \dfrac{(x + 2)(x + 3)}{-2(x + 6)}, x \neq -6, 2, x ∈ \mathbb{R}

1.40mins
Multiplying Rational Expressions

## Restriction on Division of Rational Expressions

 \displaystyle \frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C} = \frac{AB}{BC} 

where  \displaystyle B, C, D =\neq 0 

2.50mins
Restriction on Division of Rational Expressions
Sum of Rational Expressions 6 Videos

## A soft intro to Sum of Rational Expressions

A overview Introduction

## Introduction to Sum of Rational Expressions

\dfrac{A}{B} + \dfrac{C}{D} = \dfrac{AD + BC}{BD}

ex Combine and simplify

\dfrac{1}{x -1} + \dfrac{1}{2x + 1}

= \dfrac{2x + 1 + x - 1}{(x - 1)(2x + 1)}

= \dfrac{3x}{(x - 1)(2x + 1)}

2.17mins
1 Introduction to Sum of Rational Expressions

## Adding Rational Expression with more than one factors in the denominator

\dfrac{A}{BC} + \dfrac{D}{BE} = \dfrac{AE + CD}{BCE}

ex Combine and simplify.

\dfrac{1}{x^2 - 1} + \dfrac{1}{x^2 - x -2}

= \dfrac{2x - 3}{(x - 1)(x + 1)(x - 2)}

4.02mins
2 Adding Rational Expression with more than one factors in the denominator

## Subtraction of Rational Expressions

ex Subtract and simplify.

\dfrac{1}{(x -1)(x + 2)^2} - \dfrac{1}{(x -2)(x + 2)(x - 1)}

= \dfrac{-4}{(x - 1)(x + 2)^2(x - 2)}

1.39mins
3 Subtraction of Rational Expressions

## Addition of Two Rational Terms

ex Add and simplify.

\dfrac{2}{(x^2 -4)(x^2-4x +3)} + \frac{1}{(x^2 -x -2)(x^2 + 4x + 4)}

= \dfrac{3x^2 + 2x + 7}{(x-2)(x + 2)^2(x-1)(x + 1)(x-3)}

2.35mins
4 Addition of Two Rational Terms

## Adding Three Rational Terms

ex Find the sum.

\dfrac{1}{x - 2} + \dfrac{1}{(x - 2)^2} + \dfrac{1}{(x - 2)^3}

= \dfrac{x^2 - 3x + 3}{(x - 2)^3}